Question 1

Find a Maclaurin series expansion for $f(x)=\frac{e^{x}-1}{x}$.

Accepted Answer

$f(x)=1+\frac{x}{2 !}+\frac{x^{2}}{3 !}+\frac{x^{3}}{4 !}+\ldots$

Question 2

Find the Fourier series expansion of $f(x)=3 x, 0 \leq x<2 \pi$. Write at least four terms.

Accepted Answer

$f(x)=3 \pi-6 \sin x-3 \sin 2 x-2 \sin 3 x-\frac{3}{2} \sin 4 x+\ldots$

Question 3

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges.
- $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{3}+1}$

Accepted Answer

diverges by integral test

Question 4

For the problems below, determine whether each series converges or diverges.
- $1+\frac{1}{8}+\frac{1}{27}+\ldots+\frac{1}{n^{3}}+\ldots$

Accepted Answer

The answer of For the problems below, determine whether each...

Question 5

Find the Fourier series for the square wave (of period $2 \pi$ ) given by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.$

Accepted Answer

A)  $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$ 
B)  $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$ 
C)  $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$ 
D)  $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$ 
A)  $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$ 
B)  $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$ 
C)  $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$ 
D)  $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$

Question 6

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges.
- $\sum_{n=1}^{\infty} \frac{\mathrm{n}+4}{\mathrm{n} \cdot 5^{\mathrm{n}}}$

Accepted Answer

The answer of For the problems below, use either the...

Question 7

Investigate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$.

Accepted Answer

A)  The p- series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ converges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges absolutely. 
B)  The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges conditionally. 
C)  The ratio test gives $\mathrm{r}=1$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{(-1)^{\mathrm{n}+1}}{\mathrm{n}^{2}}$ diverges. 
D)  The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ diverges. 
A)  The p- series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ converges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges absolutely. 
B)  The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges conditionally. 
C)  The ratio test gives $\mathrm{r}=1$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{(-1)^{\mathrm{n}+1}}{\mathrm{n}^{2}}$ diverges. 
D)  The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ diverges.

Question 8

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.
- $\sum_{\mathrm{n}=1}^{\infty}(-1)^{\mathrm{n}+1} \frac{1}{\sqrt[3]{\mathrm{n}}}$

Accepted Answer

converges

Question 9

Use the integral test to investigate the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$.

Accepted Answer

A)  The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$, so the series $\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}=2$. 
B)  The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ converges. 
C)  The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ diverges. 
D)  The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$, so the test fails to tell us anything about the series. 
A)  The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$, so the series $\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}=2$. 
B)  The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ converges. 
C)  The integral $\int_{1}^{\infty} \frac{1}{\mathrm{x}^{3 / 2}} \mathrm{dx}=2$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ diverges. 
D)  The integral $\int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x=2$, so the test fails to tell us anything about the series.

Question 10

Use the first four non- zero terms of the Maclaurin series for $f(x)=e^{x}$ to estimate $e^{-0.3}$.

Accepted Answer

A)  .7405 
B) .7399 
C)  .7407 
D) .7402 
A)  .7405 
B) .7399 
C)  .7407 
D) .7402

Question 11

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms.
- $f(x)=\sin 3 x$

Accepted Answer

The answer of For the problems below, find a Maclaurin...

Question 12

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.
- $\sum_{\mathrm{n}=1}^{\infty}(-1)^{\mathrm{n}+1} \frac{\mathrm{n}^{3}}{\mathrm{n}^{3}+1}$

Accepted Answer

The answer of For the problems below, determine whether each...

Question 13

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms.
- $f(x)=e^{2 x}$

Accepted Answer

The answer of For the problems below, find a Maclaurin...

Question 14

Calculate the value of $\mathrm{e}^{1.2}$ using the first four non- zero terms of a Taylor series. Round to five significant digits.

Accepted Answer

The answer of Calculate the value of $\mathrm{e}^{1.2}$ using the...

Question 15

Find a Taylor series expansion for $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ with $\mathrm{a}=\pi$.

Accepted Answer

A)  $\sum_{n=0}^{\infty} \frac{(x-\pi)^{2 n+1}}{(2 n+1) !}=(x-\pi)+\frac{(x-\pi)^{3}}{3 !}+\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}+\ldots$ 
B)  $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}) !}=1-\frac{(\mathrm{x}-\pi)^{2}}{2 !}+\frac{(\mathrm{x}-\pi)^{4}}{4 !}-\frac{(\mathrm{x}-\pi)^{6}}{6 !}+\ldots$ 
C)  $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}+1) !}=(\mathrm{x}-\pi)-\frac{(\mathrm{x}-\pi)^{3}}{3 !}+\frac{(\mathrm{x}-\pi)^{5}}{5 !}-\frac{(\mathrm{x}-\pi)^{7}}{7 !}+\ldots$ 
D)  $\sum_{n=0}^{\infty} \frac{(-1)^{n+1}(x-\pi)^{2 n+1}}{(2 n+1) !}=-(x-\pi)+\frac{(x-\pi)^{3}}{3 !}-\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}-\ldots$ 
A)  $\sum_{n=0}^{\infty} \frac{(x-\pi)^{2 n+1}}{(2 n+1) !}=(x-\pi)+\frac{(x-\pi)^{3}}{3 !}+\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}+\ldots$ 
B)  $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}) !}=1-\frac{(\mathrm{x}-\pi)^{2}}{2 !}+\frac{(\mathrm{x}-\pi)^{4}}{4 !}-\frac{(\mathrm{x}-\pi)^{6}}{6 !}+\ldots$ 
C)  $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\mathrm{x}-\pi)^{2 \mathrm{n}+1}}{(2 \mathrm{n}+1) !}=(\mathrm{x}-\pi)-\frac{(\mathrm{x}-\pi)^{3}}{3 !}+\frac{(\mathrm{x}-\pi)^{5}}{5 !}-\frac{(\mathrm{x}-\pi)^{7}}{7 !}+\ldots$ 
D)  $\sum_{n=0}^{\infty} \frac{(-1)^{n+1}(x-\pi)^{2 n+1}}{(2 n+1) !}=-(x-\pi)+\frac{(x-\pi)^{3}}{3 !}-\frac{(x-\pi)^{5}}{5 !}+\frac{(x-\pi)^{7}}{7 !}-\ldots$

Question 16

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms.
-  $f(x)=\cos x, a=\frac{\pi}{4}$

Accepted Answer

The answer of For the problems below, find the Taylor...

Question 17

Evaluate $\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}$. (Use three non-zero terms.) Round to four significant digits.

Accepted Answer

The answer of Evaluate $\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}$. (Use three...

Question 18

For the problems below, find the interval of convergences of each series.
- $\sum_{n=1}^{\infty} \frac{(6 x)^{n}}{(3 n) !}$

Accepted Answer

The answer of For the problems below, find the interval...

Question 19

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally.
-  $\sum_{n=2}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}}$

Accepted Answer

converges

Question 20

Find a Maclaurin series expansion forf(x) $=e^{3 x}$.

Accepted Answer

A)  $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots$ 
B)  $\sum_{n=0}^{\infty} \frac{(3 \mathrm{x})^{n}}{n !}=1+3 \mathrm{x}+\frac{9 \mathrm{x}^{2}}{2 !}+\frac{27 \mathrm{x}^{3}}{3 !}+\ldots$ 
C)  $\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{(3 n) !}=1+\frac{3 x}{3 !}+\frac{9 x^{2}}{6 !}+\frac{27 x^{3}}{9 !}+\ldots$ 
D)  $\sum_{n=0}^{\infty} \frac{3 x^{n}}{n !}=3+3 x+\frac{3 x^{2}}{2 !}+\frac{3 x^{3}}{3 !}+\ldots$ 
A)  $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots$ 
B)  $\sum_{n=0}^{\infty} \frac{(3 \mathrm{x})^{n}}{n !}=1+3 \mathrm{x}+\frac{9 \mathrm{x}^{2}}{2 !}+\frac{27 \mathrm{x}^{3}}{3 !}+\ldots$ 
C)  $\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{(3 n) !}=1+\frac{3 x}{3 !}+\frac{9 x^{2}}{6 !}+\frac{27 x^{3}}{9 !}+\ldots$ 
D)  $\sum_{n=0}^{\infty} \frac{3 x^{n}}{n !}=3+3 x+\frac{3 x^{2}}{2 !}+\frac{3 x^{3}}{3 !}+\ldots$

Find a Maclaurin series expansion for $f(x)=\frac{e^{x}-1}{x}$ .

Find the Fourier series expansion of $f(x)=3 x, 0 \leq x<2 \pi$ . Write at least four terms.

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{3}+1}$

For the problems below, determine whether each series converges or diverges. - $1+\frac{1}{8}+\frac{1}{27}+\ldots+\frac{1}{n^{3}}+\ldots$

Find the Fourier series for the square wave (of period $2 \pi$ ) given by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \\ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.$

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - $\sum_{n=1}^{\infty} \frac{\mathrm{n}+4}{\mathrm{n} \cdot 5^{\mathrm{n}}}$

Investigate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ .

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally. - $\sum_{\mathrm{n}=1}^{\infty}(-1)^{\mathrm{n}+1} \frac{1}{\sqrt[3]{\mathrm{n}}}$

Use the integral test to investigate the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ .

Use the first four non- zero terms of the Maclaurin series for $f(x)=e^{x}$ to estimate $e^{-0.3}$ .

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - $f(x)=\sin 3 x$

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally. - $\sum_{\mathrm{n}=1}^{\infty}(-1)^{\mathrm{n}+1} \frac{\mathrm{n}^{3}}{\mathrm{n}^{3}+1}$

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - $f(x)=e^{2 x}$

Calculate the value of $\mathrm{e}^{1.2}$ using the first four non- zero terms of a Taylor series. Round to five significant digits.

Find a Taylor series expansion for $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ with $\mathrm{a}=\pi$ .

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms. - $f(x)=\cos x, a=\frac{\pi}{4}$

Evaluate $\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}$ . (Use three non-zero terms.) Round to four significant digits.

For the problems below, find the interval of convergences of each series. - $\sum_{n=1}^{\infty} \frac{(6 x)^{n}}{(3 n) !}$

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally. - $\sum_{n=2}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}}$

Find a Maclaurin series expansion forf(x) $=e^{3 x}$ .

Analytic Geometry

The Derivative

Applications of the Derivative

Derivatives of Transcendental Functions

The Integral

Applications of Integration

Methods of Integration

Three-Space: Partial Derivatives and Double Integrals

Progressions and the Binomial Theorem

First-Order Differential Equations

Second-Order Differential Equations

Filters

Exam 10: Series

Find a Maclaurin series expansion for f(x)=ex−1xf(x)=\frac{e^{x}-1}{x}f(x)=xex−1​ .

Find the Fourier series expansion of f(x)=3x,0≤x<2πf(x)=3 x, 0 \leq x<2 \pif(x)=3x,0≤x<2π . Write at least four terms.

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - ∑n=1∞n2n3+1\sum_{n=1}^{\infty} \frac{n^{2}}{n^{3}+1}∑n=1∞​n3+1n2​

For the problems below, determine whether each series converges or diverges. - 1+18+127+…+1n3+…1+\frac{1}{8}+\frac{1}{27}+\ldots+\frac{1}{n^{3}}+\ldots1+81​+271​+…+n31​+…

Find the Fourier series for the square wave (of period 2π2 \pi2π ) given by f(x)={−1,−π≤x<01,0≤x≤π\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \\ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.f(x)={−1,1,​−π≤x<00≤x≤π​

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - ∑n=1∞n+4n⋅5n\sum_{n=1}^{\infty} \frac{\mathrm{n}+4}{\mathrm{n} \cdot 5^{\mathrm{n}}}∑n=1∞​n⋅5nn+4​

Investigate the alternating series ∑n=1∞(−1)n+1n2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}∑n=1∞​n2(−1)n+1​ .

Use the integral test to investigate the series ∑n=1∞1n3/2\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}∑n=1∞​n3/21​ .

Use the first four non- zero terms of the Maclaurin series for f(x)=exf(x)=e^{x}f(x)=ex to estimate e−0.3e^{-0.3}e−0.3 .

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - f(x)=sin⁡3xf(x)=\sin 3 xf(x)=sin3x

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - f(x)=e2xf(x)=e^{2 x}f(x)=e2x

Calculate the value of e1.2\mathrm{e}^{1.2}e1.2 using the first four non- zero terms of a Taylor series. Round to five significant digits.

Find a Taylor series expansion for f(x)=sin⁡x\mathrm{f}(\mathrm{x})=\sin \mathrm{x}f(x)=sinx with a=π\mathrm{a}=\pia=π .

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms. - f(x)=cos⁡x,a=π4f(x)=\cos x, a=\frac{\pi}{4}f(x)=cosx,a=4π​

Evaluate ∫01cos⁡xdx\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}∫01​cosx​dx . (Use three non-zero terms.) Round to four significant digits.

For the problems below, find the interval of convergences of each series. - ∑n=1∞(6x)n(3n)!\sum_{n=1}^{\infty} \frac{(6 x)^{n}}{(3 n) !}∑n=1∞​(3n)!(6x)n​

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally. - ∑n=2∞(−1)nsin⁡nn2\sum_{n=2}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}}∑n=2∞​(−1)nn2sinn​

Find a Maclaurin series expansion forf(x) =e3x=e^{3 x}=e3x .

Analytic Geometry

The Derivative

Applications of the Derivative

Derivatives of Transcendental Functions

The Integral

Applications of Integration

Methods of Integration

Three-Space: Partial Derivatives and Double Integrals

Progressions and the Binomial Theorem

First-Order Differential Equations

Second-Order Differential Equations

Filters

Find a Maclaurin series expansion for $f(x)=\frac{e^{x}-1}{x}$ .

Find the Fourier series expansion of $f(x)=3 x, 0 \leq x<2 \pi$ . Write at least four terms.

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{3}+1}$

For the problems below, determine whether each series converges or diverges. - $1+\frac{1}{8}+\frac{1}{27}+\ldots+\frac{1}{n^{3}}+\ldots$

Find the Fourier series for the square wave (of period $2 \pi$ ) given by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \\ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.$

For the problems below, use either the ratio test or the integral test to determine whether each series converges or diverges. - $\sum_{n=1}^{\infty} \frac{\mathrm{n}+4}{\mathrm{n} \cdot 5^{\mathrm{n}}}$

Investigate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ .

Use the integral test to investigate the series $\sum_{\mathrm{n}=1}^{\infty} \frac{1}{\mathrm{n}^{3 / 2}}$ .

Use the first four non- zero terms of the Maclaurin series for $f(x)=e^{x}$ to estimate $e^{-0.3}$ .

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - $f(x)=\sin 3 x$

For the problems below, find a Maclaurin series expansion for each function. Include at least thereetarms. - $f(x)=e^{2 x}$

Calculate the value of $\mathrm{e}^{1.2}$ using the first four non- zero terms of a Taylor series. Round to five significant digits.

Find a Taylor series expansion for $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ with $\mathrm{a}=\pi$ .

For the problems below, find the Taylor series expansion for each function for the given value of a. Give at least three terms. - $f(x)=\cos x, a=\frac{\pi}{4}$

Evaluate $\int_{0}^{1} \cos \sqrt{\mathrm{x}} \mathrm{dx}$ . (Use three non-zero terms.) Round to four significant digits.

For the problems below, find the interval of convergences of each series. - $\sum_{n=1}^{\infty} \frac{(6 x)^{n}}{(3 n) !}$

For the problems below, determine whether each alternating series converges or diverges. If it converges, find whether it converges absolutely or converges conditionally. - $\sum_{n=2}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}}$

Find a Maclaurin series expansion forf(x) $=e^{3 x}$ .